How Do You Calculate Velocity Ratio in Gear- and Belt-Driven Systems?

Calculate velocity ratio (VR) by dividing input speed by output speed. For gears, divide the driven gear's teeth by the driver gear's teeth; for belt drives, divide the driven pulley's diameter by the driver pulley's diameter.

$$\text{VR}=\frac{\text{input speed}}{\text{output speed}}$$

The Reasoning

The driver receives the input motion, while the driven component provides the output motion. Velocity ratio is dimensionless because it compares quantities with the same units.

For meshing gears, equal motion at the contact point gives:

$$N_{\text{driver}}T_{\text{driver}}=N_{\text{driven}}T_{\text{driven}}$$

Here, $N$ is rotational speed and $T$ is the number of teeth. A 20-tooth driver turning a 60-tooth driven gear has:

$$\text{VR}=\frac{60}{20}=3$$

Therefore, the driven gear rotates at one-third of the driver's speed. If the driver rotates at $300\text{ rpm}$, the output speed is $300/3=100\text{ rpm}$.

For belt drives, equal belt speed means $D_{\text{driver}}N_{\text{driver}}=D_{\text{driven}}N_{\text{driven}}$. A driver and driven pulley produce , so an input of gives output.

Direction must be considered separately from VR magnitude.

For compound systems, multiply the individual velocity ratios. Idler gears change rotational direction but not the overall velocity-ratio magnitude. These calculations are theoretical: belt slip, friction, and deformation can reduce actual output speed. Do not confuse velocity ratio with mechanical efficiency, which compares useful output energy or power with input.

IB Exam Technique

In B3.3 Mechanical systems application and selection, label the driver and driven components, state the formula, substitute values with units, and interpret the result. A common misconception is reversing the fraction: under this convention, $\text{VR}>1$ indicates speed reduction, while $\text{VR}<1$ indicates speed increase.